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**Instrumental weighted variables under heteroscedasticity. I: Consistency.**
*(English)*
Zbl 1413.62091

Summary: The proof of consistency of the instrumental weighted variables, the robust version of the classical instrumental variables is given. It is proved that all solutions of the corresponding normal equations are contained, with high probability, in a ball, the radius of which can be selected – asymptotically – arbitrarily small. Then also \(\sqrt{n}\)-consistency is proved. An extended numerical study (the Part II of the paper) offers a picture of behavior of the estimator for finite samples under various types and levels of contamination as well as various extent of heteroscedasticity. The estimator in question is compared with two other estimators of the type of “robust instrumental variables” and the results indicate that our estimator gives comparatively good results and for some situations it is better.

The discussion on a way of selecting the weights is also offered. The conclusions show the resemblance of our estimator with the \(M\)-estimator with Hampel’s \(\psi\)-function. The difference is that our estimator does not need the studentization of residuals (which is not a simple task) to be scale- and regression-equivariant while the \(M\)-estimator does. So the paper demonstrates that we can directly compute – moreover by a quick algorithm (reliable and reasonably quick even for tens of thousands of observations) – the scale- and the regression-equivariant estimate of regression coefficients.

For Part II see [Kybernetika 53, No. 1, 26–58 (2017; Zbl 1413.62092)].

The discussion on a way of selecting the weights is also offered. The conclusions show the resemblance of our estimator with the \(M\)-estimator with Hampel’s \(\psi\)-function. The difference is that our estimator does not need the studentization of residuals (which is not a simple task) to be scale- and regression-equivariant while the \(M\)-estimator does. So the paper demonstrates that we can directly compute – moreover by a quick algorithm (reliable and reasonably quick even for tens of thousands of observations) – the scale- and the regression-equivariant estimate of regression coefficients.

For Part II see [Kybernetika 53, No. 1, 26–58 (2017; Zbl 1413.62092)].

### MSC:

62J02 | General nonlinear regression |

62G30 | Order statistics; empirical distribution functions |

62G35 | Nonparametric robustness |